Multicriteria decision making attributes and estimation of physicochemical properties of kidney cancer drugs via topological descriptors

Based on topological descriptors, QSPR analysis is an incredibly helpful statistical method for examining many physical and chemical properties of compounds without demanding costly and time-consuming laboratory tests. Firstly, we discuss and provide research on kidney cancer drugs using topological indices and done partition of the edges of kidney cancer drugs which are based on the degree. Secondly, we examine the attributes of nineteen drugs casodex, eligard, mitoxanrone, rubraca, and zoladex, etc and among others, using linear QSPR model. The study in the article not only demonstrates a good correlation between TIs and physical characteristics with the QSPR model being the most suitable for predicting complexity, enthalpy, molar refractivity, and other factors and a best-fit model is attained in this study. This theoretical approach might benefit chemists and professionals in the pharmaceutical industry to forecast the characteristics of kidney cancer therapies. This leads towards new opportunities to paved the way for drug discovery and the formation of efficient and suitable treatment options in therapeutic targeting. We also employed multicriteria decision making techniques like COPRAS and PROMETHEE-II for ranking of said disease treatment drugs and physicochemical characteristics.


Introduction
Healthy kidney cells grow out of control, resulting in renal cortical tumors and kidney cancer.It may be benign, slow-growing, or malignant.A malignant tumor is cancerous, which means it has the potential to develop and spread to other bodily regions.Although an indolent tumor can sometimes be malignant, it seldom metastasizes to other bodily regions.Kidney cancer develops in the renal tubule and collecting tubular epithelial system, which makes about 5% of all tumors.Major risk factors for kidney cancer include obesity, diabetes, hypertension, smoking, renal damage, and medication use.The main signs of kidney cancer were hematuria, renal discomfort, and mass.The disease's early stages do not have any visible symptoms.As a result, people who want to receive medical treatment may already have kidney cancer that has spread to other parts of their bodies and may be experiencing related complications [1].Malignant kidney tumors make up 2% of all cancer cases worldwide and are becoming more common.In the US, kidney cancer will see 63,000 new cases in 2018, 15,000 fatalities from the disease, and 350,000 new cases worldwide [2].The condition known as kidney cancer is made up of a variety of distinct cancers, each of which has a unique histology, clinical course, response to treatment, and underlying genetic cause [3].Over the previous 65 years, the rate of RCC has risen by 2% a year.This increase's cause is not known.RCC makes up around 90% of renal cancers, while clear cell tumors make up 85% of these.Bellini (collecting) duct tumors, papillary tumors, and chromophore tumors are some additional, less frequent cell types.Less than 1% of all cases involve collecting duct cancer.The collecting duct renal cancer type medullary renal carcinoma was first mentioned in sickle cell trait-positive patients.Among the risk factors for developing RCC are smoking and obesity [4].Transitional cell carcinoma is another name for this.It is the cause of 5% to 10% of adult kidney cancer diagnoses.This particular malignancy forms in the soft tissue of the kidney, the kidney's thin capsule-like layer of connective tissue, or the surrounding fat.Surgery is typically used to treat renal sarcoma.The grade of a tumor generally refers to the extent of cell differentiation rather than the growth rate.Prompt and correct identification is crucial to improve the treatment of patients with kidney disease, reduce excess morbidity and death, limit the overuse of antimalarial drugs, and lessen the emergence of antimalarial drug resistance.Good diagnostic techniques must be employed in wealthy countries where there is usually a knowledge gap in kidney cancer and in resource-poor locations where malaria is a severe hazard to society.The mitigation or extinction of species is a problem for the scientific community.The illness claims a lot of lives on a global basis.New pharmaceuticals are developed and studied by scientists, and their discovery is problematic because it is expensive, time-consuming, and challenging in drug discovery.Numerous judgments are executed to extravagance, and stop said disease; for this, drug tests are carryout to combat the disease.It requires early verdicts and medication that will be suitable for the condition.The study of graphs with points and lines visually illustrating mathematical facts is a branch of mathematics and computer science.It can be used in a variety of situations.The application of graph theory has grown swiftly.Thoughtful the computer's computation flow, communication networks, and data management are useful.The design of electrical connections, linguistics' parsing of language trees, the syntax of a language tree, phonology, morphology, chemistry, physics, mathematics, and biology all depend heavily on graphs.The growth of theoretical chemistry relies heavily on graph theory.Topological indices (TIs) are the numerical representations of a molecular structure that the molecular graph provides.We may learn much about a molecule's physicochemical and biological properties by using them in structure-property (QSPR) and structure-activity relationship (QSAR) investigations.In this research, potential antimalarial compounds, including valrubicin, casodex, docetaxel, darolutamide, degarelix, eligard, zytiga, erleada, flutamide, nubeqa, olaparib, mitoxanrone, nilutamide, pluvicto, relugolix, rubraca, abiraterone, zoladex and enzalutamide and other compounds are examined with TIs and linear regression modeling.Also, these medicines are harmless and operational, which is mandatory for the community.Figs 1-3 displays the drugs molecular structure.Some degree-based topological indices are built for these drugs' chemical graphs.
Additionally, some of the physicochemical features of the drugs are estimated using TIs with the aid of QSPR facsimiles.Regression technique-acquired QSPR studies may aid in creating fresh medicines for the treatment of Kidney cancer.These indices and the antimalarial activity were discovered to be related.Other compound series do very well with this relationship.Therefore, the TIs could be vital in creating and synthesizing new drugs.Still, they must first be empirically validated, as this work's conclusion demonstrates that both have a good relation.The ABC, Geometric, Zagreb, and Randic indices can be used to predict drug properties and bioactivity.In this article, we intended degree-based TIs on kidney cancer drugs.The said cancer drugs are carefully probed using TIs and imposed QSPR.It is found in this study that the regression model (RM) and TIs of drug properties have a good relationship.Through the analysis of topological indices, it is possible to discover molecular structure features in pharmaceutical sciences, which are crucial for creating novel products.In this study, the structure of a drug  is represented as a graph so that vertex V(G) expresses an atom, and each edge E(G) represents a chemical connection between these atoms.All graphs are assumed to be simple and linked.The number of edges that connect a vertex to other edges determines its degree [5].

Material and methods
In this study, kidney cancer drugs are modeled by simple graphs.To compute the topological indices of the considered drug's structure, the employed methods are vertex partitioning, edge partitioning, and computational techniques.In this study, we solely consider finite, simple, linked graphs.G is a graph with the vertex set V and the edge set E, assuming.The degree d u of a vertex u depends on how many vertices are nearby.The list of topological formulas used in this study is provided below.
Zagreb index [6] is Randic index [7] is The sum connectivity index [8] is ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi The geometric arithmetic index [9] is Redefined third Zagreb index [10] is The harmonic index [20] of G is ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffiffi Symmetric division [13] is As an outcome, the RM model is preeminent to check and use for the examination above.Recently, Khan et al. [14] discussed bladder cancer drugs and applied a QSPR model to predict several disease drug properties.Previous research on covid-19 is deliberated by Colakoglu [15].Havare discussed cancer treatment [16] and conversed that drug detection is an inflated and composite marvel that is best expected with this process.QSPR model of blood cancer drugs were done by Nasir et al. [17], which depicts the best-fit model.Developments in QSPR inspired us to work on existing research.This study aims to probe QSPR modeling of kidney disease drug regimens used in therapeutic management.The joint Rheumatoid arthritis (RA) investigation was done by Parveen et al. [18].This encouraged us to study said cancer drug QSPR's relation with TIs.Numerous experiments have discovered a clear connection between the molecular structures of chemical compounds and medications and their chemical properties, such as their boiling and melting points.Gao et al. [19] focused on a family of smart polymers frequently employed in creating anti-cancer medications.The results compensate for the lack of chemical and medical testing and serve as a theoretical framework for pharmaceutical engineering by determining the number of topological indices.The QSPR modeling of antituberculosis drugs is detailed in [20] completed the QSPR study of octane isomers and found a best-fit model for it.Earlier studies on covid-19, blood cancer, anti-cancer, and QSPR different TIs for various chemical structures motivate us much.Recently QSPR Analysis of varying skin cancer drugs are presented by Khan et al. in [21].Sultana [22] probe into infertility drugs QSPR modeling in well versed way and made valuable contribution.
The mathematical description of Metal-Organic Networks (MONs) which are used in drug delivery, catalysis and sensing is provided [23].In a number of scientific and technological domains fresh prospects are provided by a comprehensive closed mathematical description of graphyne and Zigzag graphyne nanoribbon [24].Using the Laplacian polynomial, massive graphs with millions of vertices and edges are examined [25] and find modified results.The formulas for triangular cacti, boron nanotubes and their derivatives [26] as well as octagonal cell networks were obtained [27].Numerous implications exist for Fuchsine acid in histology.QSPR evaluations were found [28] and prove elegant formulas.Reverse entropies in the critical volume and temperature analysis Henry's law are substantial and can be calculated using the QSPR model [29].For cerium oxide modified versions of the Hyper, Randić and Forgotten topological indices are examined [30].The usefulness of Ve degree indices is covered [31] while SiO 4 network's temperature index is investigated [32] so as several effective predictors are defibrated [29] and insightful findings are made.

Mathematical computation of topological indices
This section represents TIs of kidney cancer drug and the QSPR modeling on drug molecular structures.

Computation of topological indices
Let G is a graph of mitoxantrone with edge partitions.
. By applying definitions 2.1 to 2.8, we get as follows: ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi ffi Topological indices of other drugs can be calculated using the identical technique with Eqs (1) to (9) given in Section 2. Table 1 mentions the TIs, and Figs 1-3 shows drugs and these may be found at Chemicalbook.

QSPR analysis
In this section, TIs are deliberated on kidney drugs.Calculation between QSPR and TIs shows that they are correlated.The potential drugs in Table 1 are used in a mathematical analysis that is used in kidney treatment.In this article, nine TIs for QSPR modeling drive are done.Properties such as molar refractivity (R), polar surface area (PSA), polarity (P), complexity (C), Surface Tension (ST), enthalpy (E), molar volume (MV), boiling point (BP), and flash point (FP) for nine cures used for kidney usage.The regression model (RM) imposed on drugs is tested by applying the equation.Using (10) the subsequent diverse linear models is helpful to find out other TIs and given below: P is the property of the given drug.The X = TI, α is constant, and β is the regression coefficient.MATLAB and R-Language are used in the calculation.Nine TIs are used to analyze with the aid of linear QSPR.Eq (10) is relevant for results.

Linear regression models
In this section using (1), following linear regression models of TIs are mentioned Tables 2-10 shows parameters and QSPR models with TIs.

Discussion
The main objective of this section is to establish a QSPR analysis between a number of TIs and to investigate a number of physicochemical properties and activities of medications, including valrubicin, casodex, docetaxel, darolutamide, degarelix, eligard, zytiga, erleada, flutamide, olaparib, mitoxantrone, nilutamide, pluvicto, nubeqa, relugolix, rubraca, abiraterone, zoladex and enzalutamide and probed effectiveness of these TIs.The numerical values of seven physicochemical characteristics are studied, including enthalpy (EP), flash point (FP), molecular volume (Mv), topological polar surface area (PSA), complexity (C), flash point (F), and boiling point (BP) etc.We obtained these values from PubChem.Table 4 lists the values of the correlation coefficients r between the physicochemical attributes and the defined degree-based topological indices.Also, the Tables 4-10 shows that there exist a best fit linear QSPR model and helpful to predict the properties.with model and TIs such as SDD (G) deliver maximum correlation of molar PSA r = 0.941 and complexity r = 0.986.GA (G), RA (G), S (G), and H (G) have r = 0.987 maximum refractivity.ABC (G), RA (G), and S (G) deliver maximum correlation of molar polarity r = 0.986.There exists a strong correlation except in surface tension and density.

Multicriteria decision making attributes
This section uses a variety of topological indices to conduct a behavioral analysis of chemical structures.Chemical invariants are intended to give analysts and scientists an improved way of determining the chemical and physical properties of kidney cancer medications.Right now, we provide a weighted analysis that employs numerous topological indices.This weighted evaluation is carried out employing two alternative decision-making processes: Complex Proportional Assessment (COPRAS) and PROMETHEE (Preference Ranking Organization Method for Enrichment Evaluation) [33].When many criteria are involved, COPRAS is a method presented by Zavadskas, Kaklauskas, and Sarka [34] for ranking and selecting options.While faced with a dearth of factors to consider, the PROMETHEE technique [35] offers helpful guidance as you make decisions.PROMETHEE II provides a thorough ranking, in contrast to PROMETHEE I, which only provides a partial ranking.The aforementioned improved technique is characterized by the control of incomparability throughout the entire ranking and the removal of scale impacts across criteria.Because it has a major impact on evaluation findings, the weight design is a crucial component of MCDM approaches.The weight of each criterion in this work is determined using the Entropy Method.To determine the weights of the criterion, the problem can be handled by applying the entropy technique.Information theory has incorporated it ever since its inception in the field of thermodynamics [36].It can determine how much important information there is based on the data that is currently available.When items with the same indicator but a low entropy show a large variance, the indicator with the lower entropy provides more significant information, and the weight needs to be adjusted accordingly.The relative weight will decrease if the entropy increases and the discrepancy decreases.Thus, the entropy technique can be used to objectively measure weight [37].

COPRAS attribute
The well-known MCDM technique known as Complex Proportional Assessment (COPRAS) is applied in numerous disciplines [38].The previously mentioned method determines the best and worst options from a range of possibilities.The main goal of the COPRAS technique  is to assign multiple criteria to each alternative in order to rate them.The Tables 11-13 contains computed data for implementing COPRAS attribute and Y i and Qu i are computed by using following equations.

PROMETHEE-II
Brans invented the PROMETHEE technique, which includes PROMETHEE II for whole ranking of alternatives and PROMETHEE I for partial ranking [39].In comparison to other multicriteria analysis systems, this ranking method is extremely simple to design and use.The purpose of this is to assess and select a small number of options based on one or more criteria [40].PRO-METHEE requires two additional details.(i) Details regarding the level of relative importance, including the weights assigned to specific criteria.and (ii) Data pertaining to the comparative significance of each criterion, particularly the allocated weights.The information concerns decision-makers who are accustomed to independently evaluating the benefits of each choice in relation to every criterion.The weighted coefficients might be determined via the entropy method.We considered PROMETHEE II since we desired a detailed ranking of the possibilities.
For the beneficial criteria max l ij À min l ij for j 2 1; 2; . . .; n f g and i 2 1; 2; . . .; m f g For the non-beneficial criteria max l ij À min l ij for j 2 1; 2; . . .; n f g and i 2 1; 2; . . .; m f g Tables 14 and 15 contain computational data related to PROMETHEE II.

Conclusion
In this study, we estimated TIs for medications used to treat kidney cancer and compared them to the QSPR model.Theoretical inferences from this article's findings are advantageous for developing new kidney cancer drugs.Our findings reveal a distinct trend in investigating structures and their physical characteristics.Finally, the study helps to design new kidney drugs efficiently and preventive measures for the disease mentioned above.QSPR and TI's premeditated tenets are eye-opening to chemists deployed on drug-determining phenomena in the pharmaceutical industry.They provide new approaches in estimating properties for specific diseases and drugs, but it is essentially validated, as this work's conclusion demonstrates.In this study, we also employed two

Fig 4 .
Fig 4. Correlation coefficient and TIs.https://doi.org/10.1371/journal.pone.0302276.g004 multicriteria decision making techniques like COPRAS and PROMETHEE-II to make ranking of the medicines used for kidney cancer treatment as shown in Figs5 and 6, respectively.

Table 8 .
Regression model of H (G).

Table 13 . Quantitative utility and ranking.
1. ABC, RA and S index gave best results of molar volume with r value 0.986.